5. Consider the mapping w =-. (a) Find the image of the line Im(a)- -1 under thi

Question

5. Consider the mapping w =-. (a) Find the image of the line Im(a)- -1 under this transformation. 5 points (b) Show that this imge s the crke 5 points 6. Explain the difference between "f(z) is diferentiable at : = to" and "f(z) is analytic at 5 points 7. Use limits to show that f(e) - Re(e) is not differentiable at : -1+2i. 10 points s. (a) Describe the Canchy-Riemann equations o points (b) Let u(r,u)ry. Find all possible functions tv(r. u) such that u + ir is entire. 10 points 9. Graduate Students Only : Let w be any principal 5-1h root of unity, ie·ws-i, w. Show that 1 +2+2+2+/-o 10 points

Explanation / Answer

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6. There is a slight difference between being differentiable at a point and being analytic at a point.
Differentiable-

In the complex plane a function is said to be differentiable at a point z0z0 if the limit

limz->z0 f(z)f(z0)/(zz0)

exists. Then the limit is called the complex derivative of the function f at z0

Analytic:
An already differentiable function is said to be analaytic at z0 , if its differentiable at z0 and its neighbourhood, essentially it needs to have infinite differentials in the neighbourhood
technically,
If U is an open set, i.e every point in U is an interior point, and f is complex differentiable at every point in U, then f is said to be analytic on U

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